I'm trying to find a standard name or standard reference for two simple-to-prove relations involving binomial distributions.


$b(n,r,p) := \binom{n}{r}p^r(1 - p)^{n-r}$

i.e., it is the probability that, out of n coin tosses, exactly r turn up heads if the probability of a heads in each coin toss is p.

Then, the first identity is:

$b(n,r,p) = pb(n-1,r-1,p) + (1 - p)b(n-1,r,p)$

This can be proved directly using probabilistic arguments, or through algebraic simplication combined with the use of Pascal's identity. I thought it should be called the "probabilistic Pascal's identity", but Google doesn't agree with me.

The second identity for which I'm looking for a name is:

$\frac{\partial}{\partial p} b(n,r,p) = n(b(n-1,r-1,p) - b(n-1,r,p))$

This is easy to prove algebraically, using another recurrence relation for the binomial coefficients along the way; it probably also has a probabilistic proof.

These identities came up in the proofs of some theorems in an economics paper (models of utility functions of discrete variables) and the person writing the paper asked me if there are standard names/references for the identities, so that they can be referred to using those names without having to do the algebra in the paper.

Also, if anybody knows a place where one can enter an identity and determine whether it has a standard name or reference, that would be great.


I don't think either of these have names. If they did, they're not standard names. So unless you really need to keep the paper short, I'd just include the proofs. (They're simple enough that I don't think they'd be out of place in an economics paper.)


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